Common divisors of totients of polynomial sequences

Motivated by a question of Venkataramana, we consider the greatest common divisor $\phi(f(n))$ where $f$ is primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, establish that this gcd may be bounded just in terms degree $f$. Unconditionally such bound for quadratic polynomials, as well polynomials split completely into linear factors. The paper also addresses Calegari, establishes there are infinitely many $n^2+1$ not divisible any prime $\equiv 1 \bmod 2^m$ provided $m$ large fixed integer.